Zero Polynomial problem

Let have degree and let with degree of at least 1 with the leading coefficient of B being a unit of R. Then by a problem I posted earlier, we know that there exist unique polynomials such that . Prove that if , then is not the zero polynomial.

Idea so far.

So if the degree of B is 1, then the degree of is less than 1, so does that force it to be a constant polynomial that is not 0 since the leading coefficient is a unit?